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Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange equation of motion for this action is $\nabla_{\dot{\gamma}}\dot{\gamma}=0,$ where $\nabla$ is the Levi-Civita connection. This equation makes sense of course for any connection, so we may define geodesics with respect to that connection. Is there some functional on paths which depends only on the connection and yields this EOM?

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What kind of connection are you interested in -- how general? Just linear/affine, or something more general than that? –  Ryan Budney Oct 25 '12 at 21:02
    
@RyanBudney Which ones have action functionals? –  Joe Hannon Oct 26 '12 at 21:05

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